Theory of "Weak Value" and Quantum Mechanical Measurements

We review the definition and the concepts of the weak values and some measurement model to extract the weak value. This material is based on the author Ph.D. thesis "Time in Weak Values and Discrete Time Quantum Walk" at Tokyo Institute of Technology (2011).Comment: to be published from "Measurements in Quantum Mechanics", edited by M. R. Pahlavani (InTech, 2012) Chapter 4 page 75. Yutaka Shikano (2012). ISBN: 978-953-51-0058-4 Available from: http://www.intechopen.com/articles/show/title/theory-of-weak-value-and-quantum-mechanical-measuremen

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.Comment: This is the invited review paper on the special issue on Theoretical and Mathematical Aspects of Discrete Time Quantum Walk from Journal of Computational and Theoretical Nanoscienc

Notes on Inhomogeneous Quantum Walks

We study a class of discrete-time quantum walks with inhomogeneous coins defined in [Y. Shikano and H. Katsura, Phys. Rev. E {\bf 82}, 031122 (2010)]. We establish symmetry properties of the spectrum of the evolution operator, which resembles the Hofstadter butterfly.Comment: 4 pages, 2 figures, as the proceedings of QCMC 201

Stereographical visualization of a polarization state using weak measurements with an optical-vortex beam

We propose a stereographical-visualization scheme for a polarization state by two-dimensional imaging of a weak value with a single setup. The key idea is to employ Laguerre-Gaussian modes or an optical vortex beam for a probe state in weak measurement. Our scheme has the advantage that we can extract information on the polarization state from the single image in which the zero-intensity point of the optical vortex beam corresponds to a stereographic projection point of the Poincare sphere. We experimentally perform single-setup weak measurement to validate the stereographical relationship between the polarization state on the Poincare sphere and the location of the zero-intensity point.Comment: 6 pages, 4 figure